Joint time-state generalized semiconcavity of the value function of a jump diffusion optimal control problem

We prove generalized semiconcavity results, jointly in time and state variables, for the value function of a stochastic finite horizon optimal control problem, where the evolution of the state variable is described by a general stochastic differential equation (SDE) of jump type. Assuming that terms comprising the SDE are C-smooth, and that running and terminal costs are semiconcave in generalized sense, we show that the value function is also semiconcave in generalized sense, estimating the semiconcavity modulus of the value function in terms of smoothness and generalized semiconcavity moduli of data. Of course, these translate into analogous regularity results for (viscosity) solutions of integro-differential Hamilton–Jacobi–Bellman equations due to their controllistic interpretation. This paper may be seen as a sequel to Feleqi (Dyn Games Appl 3(4):523–536, 2013), where we dealt with the generalized semiconcavity of the value function only in the state variable. Mathematics Subject Classification. 35D10, 35E10, 60H30, 93E20.


Introduction
In this article we continue our work initiated in [26] on establishing generalized semiconcavity results for the value function of a finite horizon jump diffusion optimal control problem. While in [26] we dealt with the problem of obtaining generalized semiconcavity estimates for the value function in the state variable, uniformly in time, here we prove generalized semiconcavity results in time and state variables jointly.
Under appropriate assumptions on the data-which follow from those made in this paper-the value function can be interpreted as the unique vis-The results of this paper are a (small) part of the vast regularity theory of PIDEs. First results on this subject were obtained assuming nondegenerate diffusions or elliptic second-order differential (local) terms as in [7,28,30] (just to mention a very few references without any pretense of completeness) and references therein. Recently, there has been a revival of interest on the theory of PIDEs which is due to the work on one hand of Caffarelli et al. [13][14][15][16][17][18], and Barles, Chasseigne, Ciomaga, Imbert on the other [4][5][6]. These authors, differently from the earlier ones, prove regularity results, such as Hölder, Lipschitz, C 1,α -estimates, under a kind of ellipticity assumption, which is not any more due to the second-order local terms (or to the presence of nonsingular diffusions), but comes either from the nonlocal terms or from the combined effect of both local and nonlocal terms. Related results have been obtained by other authors as well [10,25,32,34,36,44,45,49,50].
Our interest in the regularity theory of partial integro-differential equations of HJB type and related optimal control problems arose from the recent theory of Mean Field Games (abbr. MFG) developed by Lasry and Lions [46][47][48], which yields limiting models for symmetric, non-zero sum, non-cooperative N -player games with the interaction between the players being of mean-field type. It is of interest to study MFG models where the dynamic of an average or representative agent is a jump diffusion because stochastic phenomena in Economics and Finance applications exhibit jumps and other deviations from pure diffusions. The MFG paradigm would lead in this case to PIDEs of HJB for the optimal values of the average agents coupled with Fokker-Planck PI-DEs for probability distributions of their optimal dynamics. To our knowledge the study of such systems of PIDEs remains largely to be done. In particular, we are interested in extending to these systems of PIDES our results in [3,26].
The proof is based on interpreting the said solution of (1.1) as the value function of a stochastic optimal control problem for jump diffusion processes, that is, processes which are solutions of appropriate stochastic differential equations of jump type driven by Brownian motions and Poisson random measures independent of each other (abbr. SDEs) see, e.g., [54] and references therein. Furthermore, we rely on the method of affine time changes for Brownian motions as in [11,12] and for Poisson random measures as in [33]. While the corresponding change of variable formula for Wiener integrals is rather easy, for stochastic integrals with respect to Poisson random measures, the formula is more involved and requires a change of probability on the underlying sample space via the so called Kulik's transformation; see [33] for more details and references. Other tools are Burkholder type inequalities as stated for example in [43], and of course Gronwall's inequality.
The paper is organized as follows. Main results (Theorems 2.2 and 2.11) are stated in the next section. The proof of technical lemmas is postponed to the "Appendix" (Sect. 3) in order to ensure a better readability of the paper.
Notation In accordance with common practice, we usually use the same letter (here C δ ) to denote possibly different constants in a chain of estimates/inequalities, which however depend only on the same data; see, e.g., the proofs of Lemmas 2.5 and 2.7.
that satisfy the following conditions: • is a complete filtered probability space such that the filtration F satisfies the usual hypotheses (that is, F is right continuous and every sub-σalgebra F t , for 0 ≤ t ≤ T , is complete with respect to the probability measure P; is a F-adapted Poisson random measure on R + × Z and on probability space (Ω, F, P) with intensity measure ν on Z, and with associated compensatorÑ =Ñ (dtdz) = N (dtdz) − dtν(dz); • W and N are independent of each-other and moreover have increments that are independent of the filtration F, that is, be (measurable) maps, p ≥ 2, C i , L i ≥ 0 fixed constants and ω i regularity moduli, for i = 1, . . . , 6. Assume that the following hold true: 2 Which we could call probability references if we were to adapt a terminology analogous to the one adapted in [27].
(ii) (Lipschitz continuous costs) Since p ≥ 2 and ν(Z) < ∞, then it follows that estimates for H and K hold also for p = 2. We cannot handle arbitrary moduli, therefore we have to make assumptions on the moduli as well. However, these assumptions are not very restrictive and are verified by the moduli appearing in most applications of interest. We should notice that in many cases we can replace the regularity or semiconcavity modulus of a map by a larger one so that it satisfy our assumptions. For alternative assumptions on the moduli see Theorem 2.11 below.
To begin with we make either one of the following assumptions on the moduli. (MP) (Power type moduli).
(i) (Moduli of the dynamics). We assume that for given 0 < α i (≤ 1), k i ≥ 0 and also that (ii) (Moduli of the costs). Furthermore, we assume that for given 0 < α i (≤ 1), k i ≥ 0 and also that Alternatively, we assume the following hold true.
(i) (Moduli of the dynamics). The functions (ii) (Moduli of the costs). Furthermore, the functions are concave and, if r i ≥ 1 are such that r −1 i + q −1 i = 1 for all i = 5, 6, then we assume also that Remark 2.1. The larger the p is the more restrictive these assumptions become, so we aim at proving results for p ≥ 2 as small as possible. In the case of (MP), by (2.3), (2.4), it suffices to assume that the above estimates (L), (S) hold true for p = 4. as it is done in [33], where the case of classical semiconcavity estimates (that is, ω-semiconcavity estimates with linear ω-s) is treated. Indeed, it is not reasonable to take α i > 1 (i = 1, . . . , 6), that is, a superlinear modulus, otherwise, by [21, Theorem 2.1.9], an ω-semiconcave function would just be concave and a C 1,ω map would just be constant. In such a case one may just take ω i = 0, that is, k i = 0 and α i = 0. Still we cannot assume p = 2 unless our results trivialize for this would force us to take α i = 0 for all i = 1, . . . , 4.
For any s ∈ [0, T ], R ∈ R s as in (2.1), we consider the following optimal control problem: (admissible controls) we take as set of admissible controls A R (s, T ) the set of R-predictable 3 processes α(·): [0, T ] → A; (controlled system) for any x 0 ∈ R d , α(·) ∈ A R (s, T ) we consider the stochastic differential equation of jump type: for all s ≤ t ≤ T is the solution 4 to (2.5), the cost is given by where, for each α(·) ∈ A R (s, T ), x(·) is the solution of Eq. (2.5); we consider also , R ∈ R s , and V is actually the unique viscosity solutions of (1.1) with polynomial growth [54,55]. Actually, in [33] it is proved that V is Lipschitz continuous on [0, T − δ] × R d for any δ ∈ ]0, T ].
As we pointed out in the introduction, V is not in general locally Lipschitz continuous (and therefore not semiconcave in generalized sense) . However, we prove that, for every 0 < δ ≤ T , V is ω-semiconcave on [0, T − δ] × R d for some modulus ω which can be expressed in terms of the moduli of the data of the problem.
Thus we fix also a δ ∈ ]0, T ]. We prove the following generalized semiconcavity estimates in time-space. In order to prove generalized semiconcavity estimates for V on [0, T − δ] × R d (and in particular, Theorem 2.2 above) we take s 1 , T ), and denote by τ 1 , τ 2 the affine "time changes" that transform [s 1 , T ], respectively, [s 2 , T ], into [s λ , T ], that is,

Theorem 2.2. Assume (B), (L), (S) and either (MP) or (MC). Then the value function
. We take where F i is defined as in (3.6) and Q i as in (3.9). It is easy to see that Denoting by x i (·) the solutions of Eq. (2.5) for R = R i , α(·) = α i (·) and initial conditions s = s i , x 0 = x 0 i , for i = 1, 2, respectively; and by x λ (·) the solution of (2.5) for the previously fixed R ∈ R s λ , α(·) ∈ A R (s λ , T ), initial conditions s = s λ , , we obtain, by Burkholder inequalities and change of variable formulas for stochastic integrals with respect to affine time changes-see the detailed proof in the "Appendix"-the following estimates:

Lemma 2.3. For some c ≥ 0, that depends only on d, m, T, p, ν(Z), and for every t ∈ [s λ , T ],
For a better readability of the paper, the proof of this lemma and the others stated in this section is postponed to the "Appendix".
We need the following simple technical lemma which can be checked by straightforward computation, hence its proof, which in any case can be found in [11], is omitted. provided that γ, defined by setting γ(ρ) = ρ β ω 2 (ρ) q for all ρ ≥ 0, where

By (L)-(ii), (S)-(ii)
, By Lemma 2.8 and Lemma 2.4, more specifically, (2.14), (2.18), for some constant C δ ≥ 0 that depends only on d, m, p, δ, T, ν(Z), C i , L i , i = 1, . . . , 6. Under our assumptions on the moduli (MP) or (MC) Lemma 2.7 holds true, which, in case of assumptions (MC), we apply together with Lemma 2.8, in order to deduce, by using also a final time Lemma 2.5, that (2.9). From this last estimate, since R ∈ R s λ and α(·) ∈ A R (s λ , T ) are arbitrary, it follows that V is ω-semiconcave. Remark 2.9. Up to estimate (2.2) in the proof above we do not use the assumptions on the particular form of the moduli. This is important to notice for the general estimate (2.24) may by used to obtain generalized semiconcavity estimates for other types of moduli from those envisioned in Theorem 2.2.
It should be now rather straightforward to state results under the assumption that some of the moduli ω i are of power type while the others satisfy suitable concavity properties (as stated in Lemmas 2.6 and 2.8).
In many cases of interest it is possible to choose the moduli ω i concave, and by growth assumptions contained in (B), (L), it is also possible to assume these moduli ω i bounded as well. This remark can be used to derive ω-semiconcavity results by means of the following lemma. Lemma 2.10. (bounded concave moduli) Fix q, r ∈ [1, ∞] such that 1/q+1/r = 1 and let ξ be as in Lemma 2.6 (or as in Lemma 2.8). Assume that ω q is concave for some q > 0, and ω is bounded by some constant k ≥ 0. Then Then this lemma can by used to prove the following theorem in the same fashion as we did with Theorem 2.2.
Relying on the lemmas and techniques given above, one can obtain additional results on the time-space semiconcavity of the value function, estimating, if one so wishes, the semiconcavity modulus of the value function in terms of the moduli of the data (that is, results of the type of Theorems 2.2 and 2.11) when one assumes moduli of "mixed type", that is, some moduli of power type and the others having suitable concavity properties and/or being bounded. Since the resulting statements and method of proof of these results should be clear by now, we are not providing them here. We just emphasize that, in obtaining such results, the starting point is estimate (2.24), which holds true for any moduli ω i , i = 1, . . . , 6. Then one needs to apply Lemma 2.6 and/or the first part of Lemma 2.10, firstly, to obtain a new version of Lemma 2.7 (based on the assumptions on the type of the moduli o i , i = 1, . . . , 4), and finally, one concludes by using this new version of Lemma 2.7, estimate (2.24) and/or Lemma 2.8 and/or the second part of Lemma 2.10 (whether and which of the said lemmas is to be used or not depends on the assumptions on ω i -s).

Appendix
Proof of Lemma 2.3. Fact 1. (Burkholder-Davis-Gundy inequalities [43]) For  for all predictable processes σ ∈ L 2 s i , τ −1 i (t) × Ω, dr ⊗ P; R m×d , t ∈ [s i , T ]. Next, we use a transformation of a Poisson random measure with respect to affine time changes which is called Kulik's transformation. The reader interested for more information on this transformation is referred to papers [41,42], or even [33] for a quick and very readable introduction. We define